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In circular motion, suppose $\overrightarrow{a}$ is the acceleration of particle,
$\overrightarrow{v}$ its linear velocity and $\overrightarrow{\omega }$ its angular velocity.
$$\begin{array}{cc}& \\ \text{Column-I}& \text{Column-II}\\ \text{(a)}|\overrightarrow{a}\times \overrightarrow{v}|& \text{(p) positive}\\ \text{(b)}\overrightarrow{a}.\overrightarrow{v}& \text{(q) Negative}\\ \text{(c)}\overrightarrow{\omega }.\overrightarrow{v}& \text{(r) zero}\\ \text{(d)}|\overrightarrow{\omega }\times \overrightarrow{a}|& \text{(s) Undefined}\end{array}$$

Two capacitors each of 1μF capacitance are connected in parallel and are then charged by 200V d.c. supply. The total energy of their charges (in joules) is

In self induction, ___.

A thick plane mirror shows a number of images of the filament of an electric bulb. Of these, the brightest image is the

The potential energy $U$ of a particle of mass $m=1\text{kg}$ moving in $x-y$ plane is given by $U=3x+4y,$ where $x$ and $y$ are in metre and $U$ is in joule. If initially particle was at rest, then its speed at $t=2\text{s}$ will be

A uniform chain of mass $m$ and length $l$ is lying on a frictionless horizontal table with one-third of its length hanging over the edge of the table. The amount of work done to pull the hanging part of the chain up to the table is given by

The binding energy of deutron ${}_1{H}^2$ is $1.112MeV$ per nucleon and an alpha particle ${}_{2}H{e}^4$ has a binding energy of $7.047MeV$ per nucleon. Then, in the fusion rection
${}_1{H}^2{+}_1{H}^2{\to }_{2}H{e}^4+Q$, the energy $Q$ released is
(take mass of ${}_1{H}^2=2.01478u$;
mass of ${}_{2}H{e}^4=4.00338u$)

A current of 4A flows in a inductor coil when connected to a 12 V DC source. If the same coil is connected to a 12V, 50 rad/sec AC source a current of 2.4 A flows in the circuit. If a $2500\mu F$ capacitor is connected in series to the coil the power dissipated in the circuit is

The moment of inertia of a uniform rod of mass 0.50 kg and length 1 m is 0.10 ${\text{kg-m}}^2$ about a line perpendicular to the rod. Find the distance of this line from the middle point of the rod.

Find the ratio of tension $T_1$ and $T_2$ in the string as shown in figure if the whole system is moving with constant acceleration of $4m/s^2$ under the influence of force of magnitude $220N$ acting on block of mass $15kg$ . Assume the coefficient of friction to be $\mu =1/3$ and $g=10m/s^2$.

The position of the particle is given by $x=(3t^2-e^t)$ in $\text{m}$. The velocity $\left(\text{in m/s}\right)$ of the particle as a function of time $t$ is

Why is the farad is an inconveniently large unit of capacitance.?

The initial velocity of a body moving along a straight line is $7\text{m/s}$. It has a uniform acceleration of $4{\text{m/s}}^2$. The distance covered by the body in the $5^{\text{th}}$ second of its motion is

A 10 m wide spacecraft moves through the inter steller space at a speed $3\times {10}^{7}m{s}^{-1}.$ A magnetic foeld B = $3\times {10}^{-10}$ T exicts in the space in a direction perpendicular to the plane of motion. Treating the spacecraft as a cond uctor, calculate the emf induced across its width.

Find the MOI of four identical solid spheres each of mass $M$ and radius $R$ placed in a horizontal plane as shown in figure. Line $X{X}^{\prime }$ touches two spheres and passes through the diameter of the other two spheres.

The decay constant of ${}^{238}\text{U}$ is $4.9\times {10}^{-18}{S}^{-1}$ (a) What is the average - life of ${}^{238}\text{U}$ ? (b) What is the half-life of ${}^{238}\text{U}$ ? (c) By what factor does the activity of a ${}^{238}\text{U}$ sample decrease in $9\times {10}^9$ years ?

The length of a strip measured with a meter rod is 10.0 $cm$. Its width mesasured with a vernier callipers is 1.00 $cm$. The least count of metre rod is 0.1 $cm$ and that of vernier calliipers is 0.01 $cm$. What will be the error in its area?

In figure the upper wire is made of steel and the lower of copper. The wires have equal cross section. Find the ratio of the longitudinal strains developed in the two wires.

A ball of mass 50 g moving at a speed of 2.0 m/s strikes a plane surface at an angle of incidence ${45}^{\circ }$ . The ball is reflected by the plane at equal angle of reflection with the same speed. Calculate the magnitude of the change in momentum of the ball.

A $20$ Henry inductor coil is connected to a $10ohm$ resistance in series as shown in figure. The time at which rate of dissipation of energy (joule's heat) across resistance is equal to the rate at which magnetic energy is stored in the inductor is

A block is placed on the top of a plane inclined at ${37}^{\circ }$ with horizontal. The length of the plane is 5m. The block slides down the plane and reaches the bottom. Find the speed of the block at the bottom if the coefficient of friction is 0.25.

A cell sends a current $i_1$ through a coil of resistance $R_1$ and a current $i_2$ through a coil of resistance is $R_2$.The emf of the cell is

A capacitor has capacitance of $6nF$, maximum voltage it can sustain is $40$ volt. It is in series with other capacitor $3nF$ and this capacitor can sustain $30$ volt, what is maximum voltage that can be applied in series.

The figure shown below represents two paths that are taken by an ideal gas to go from the state A to state C. In process $\text{AB},400\text{J}$ of heat is added to the system and in process $\text{BC},100\text{J}$ of heat is added to system. The heat absorbed by system in process $\text{AC}$ will be

Four identical rods of mass $M=6\text{kg}$ each are welded at their ends to form a square and then welded to a massive ring having mass $M=4\text{kg}$ and radius $R=1\text{m}$. If the system is allowed to roll down the incline of inclination $\theta ={30}^{\circ }$.


The minimum value of friction coefficient to prevent slipping is

A hemispherical bowl of radius R is rotated about its axis of symmetry which is kept vertical. A small block is kept in the bowl at a position where the radius makes an angle $\theta$ with the vertical. The block rotates with the bowl without any slipping. The friction coefficient between the block and the bowl surface is $\mu$. Find the range of the angular speed for which the block will not slip.

A bullet going with speed 320$\frac{m}{s}$ enters a concrete wall penetrates a distance of 5.0 cm before coming to rest. Find the deceleration.

A uniform rod AB of length $l$ and mass $m$ hangs from point $A$ in a car moving velocity $V_0$ on an inclined plane as shown in the figure. The rod can rotate in vertical plane about the axis at point $A$. If the car suddenly stops , the angular speed with which the rod starts rotating is

A body of mass $3\text{kg}$ is under a constant force has displacement $s$, given by the relation $s=\frac{1}{3}{t}^2$, where $t$ is in seconds, $s$ is in metres. Work done by the force in $2$ seconds is

Determine the rms value of a semi-circular current wave which has a maximum value of a Ampere. The y axis denotes current in Ampere and x axis denotes time.

The equation of a plane progressive wave is $y=0.09\text{sin}8\pi (t-\frac{x}{20})$. When it is reflected at rigid support, its amplitude become ${\frac{2}{3}}^{rd}$ of its previous value. Then the equation of the reflected wave is

A Particle is performing SHM. If its displacement from mean position is x, corresponding resultant force on it is F and acceleration is a, which of the following graphs is correct?

A man in a balloon, throws a stone downwards with a speed of $5\text{m/s}$ with respect to the balloon. The balloon is moving upwards with a constant acceleration of $5{\text{m/s}}^2$. Then velocity of the stone relative to the man after $2\text{seconds}$ is :

A thin equiconvex lens of focal length f(in air) and refractive index m has thickness t. How much thick it will appear to an observer as shown in figure?

The equation of a sound wave in air is given by

$p=\left(0.01N{m}^{-2}\right)sin\left[\right(1000s^{-1})t-(3.0m^{-1}\left)x\right]$

(a) Find the frequency, wavelength and the speed of sound wave in air.

(b) If the equilibrium pressure of air is $1.0\times {10}^{5}N{m}^{-2}$, what are the maximum and minimum pressures at a point as the wave passes through that point.

Figure shows water filled in a symmetrical container. Four pistons of equal area $A$ are used at the four openings to keep the water in equilibrium. Now an additional force $F$ is applied at each piston. The increase in the pressure at the centre of the container due to this addition is

A smooth ball A of mass is attached to one end of a light inextensible string, and is suspended from fixed point O. Another identical ball B, is dropped from a height h, so that the string just touches the surface of the sphere.

A particle starts with initial velocity $15\text{m/s}$. The $a-x$ curve is shown in the figure for the motion of particle. The speed of the particle at $x=5\text{m}$ is

Five identical rods are joined as shown in figure. Point A and C are maintained at temperature ${120}^{\circ }C$ and ${20}^{\circ }C$ respectively. The temperature of junction B will be

A ceiling fan is rotating with an angular velocity of $4rev/s$. It takes $40s$ to stop when it is switched off. The angular retardation during this interval is

The minimum energy required to launch a $m\text{kg}$ satellite from the earth’s surface in a circular orbit at an altitude $2R$, where $R$ is the radius of earth is

In determination of young modulus of elasticity of wire, a force is applied and extension is recorded. Initial length of wire is '1 m'. The curve between extension and stress is depicted then Young's modulus of wire will be K $\times$ ${10}^9$ $\frac{N}{m^2}$, where K is .

A stone is thrown at an angle $\theta$ to the horizontal reaches a maximum height $H$, then the time of flight of stone will be